Question 12.6.2: Amount of Drug Remaining in the Bloodstream To test the leng......
Amount of Drug Remaining in the Bloodstream
To test the length of time that an infection-fighting drug stays in a person’s bloodstream, a doctor gives 300 milligrams of the drug to 10 patients, labeled 1–10 in the table below. Once each hour, for 10 hours, one of the 10 patients is selected at random and that person’s blood is tested to determine the amount of the drug remaining in the bloodstream. The results are as follows.
Determine at a level of significance of 5% whether a correlation exists between the time elapsed and the amount of drug remaining.
| 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | Patient |
| 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | Time (hr) |
| 85 | 90 | 100 | 210 | 120 | 140 | 210 | 200 | 230 | 250 | Drug remaining (mg) |
Let time be represented by x and the amount of drug remaining by y. We first draw a scatter diagram (Fig. 12.42).
The scatter diagram suggests that, if a correlation exists, it will be negative. We now construct a table of values and calculate r.
\begin{aligned} r & =\frac{n(\Sigma x y)-(\Sigma x)(\Sigma y)}{\sqrt{n\left(\Sigma x^2\right)-(\Sigma x)^2} \sqrt{n\left(\Sigma y^2\right)-(\Sigma y)^2}} \\ & =\frac{10(7500)-(55)(1635)}{\sqrt{10(385)-(55)^2} \sqrt{10(302,925)-(1635)^2}} \\ & =\frac{-14,925}{\sqrt{825} \sqrt{356,025}} \approx \frac{-14,925}{17,138.28} \approx-0.871 \end{aligned}From Table 12.9, for n = 10 and a = 0.05, we get 0.632. Since \left| -0.871 \right| = 0.871 and 0.871 > 0.632, a correlation exists. The correlation is negative, which indicates that the longer the time period, the smaller is the amount of drug remaining.
| xy | y² | x² | y | x |
| 250 | 62,500 | 1 | 250 | 1 |
| 460 | 52,900 | 4 | 230 | 2 |
| 600 | 40,000 | 9 | 200 | 3 |
| 840 | 44,100 | 16 | 210 | 4 |
| 700 | 19,600 | 25 | 140 | 5 |
| 720 | 14,400 | 36 | 120 | 6 |
| 1470 | 44,100 | 49 | 210 | 7 |
| 800 | 10,000 | 64 | 100 | 8 |
| 810 | 8100 | 81 | 90 | 9 |
| 850 | 7225 | 100 | 85 | 10 |
| 7500 | 302,925 | 385 | 1635 | 55 |
| Table 12.9 Critical Values* for the Correlation Coefficient, r | ||
| n | α= 0.05 | α= 0.01 |
| 4 | 0.95 | 0.99 |
| 5 | 0.878 | 0.959 |
| 6 | 0.811 | 0.917 |
| 7 | 0.754 | 0.875 |
| 8 | 0.707 | 0.834 |
| 9 | 0.666 | 0.798 |
| 10 | 0.632 | 0.765 |
| 11 | 0.602 | 0.735 |
| 12 | 0.576 | 0.708 |
| 13 | 0.553 | 0.684 |
| 14 | 0.532 | 0.661 |
| 15 | 0.514 | 0.641 |
| 16 | 0.497 | 0.623 |
| 17 | 0.482 | 0.606 |
| 18 | 0.468 | 0.590 |
| 19 | 0.456 | 0.575 |
| 20 | 0.444 | 0.561 |
| 22 | 0.423 | 0.537 |
| 27 | 0.381 | 0.487 |
| 32 | 0.349 | 0.449 |
| 37 | 0.325 | 0.418 |
| 42 | 0.304 | 0.393 |
| 47 | 0.288 | 0.372 |
| 52 | 0.273 | 0.354 |
| 62 | 0.250 | 0.325 |
| 72 | 0.232 | 0.302 |
| 82 | 0.217 | 0.283 |
| 92 | 0.205 | 0.267 |
| 102 | 0.195 | 0.254 |
